Top person sorted by score

At this site we keep several lists of primes, most notably the list of the 5,000 largest known primes. Who found the most of these record primes? We keep separate counts for persons, projects and programs. To see these lists click on 'number' to the right.

Clearly one 100,000,000 digit prime is much harder to discover than quite a few 100,000 digit primes. Based on the usual estimates we score the top persons, provers and projects by adding ‎(log n)³ log log n‎ for each of their primes n. Click on 'score' to see these lists.

Finally, to make sense of the score values, we normalize them by dividing by the current score of the 5000th prime. See these by clicking on 'normalized score' in the table on the right.

rank person primes score 21 Michael Shafer 1 52.2829 22 Arno Lehmann 3 52.2822 23 Sylvanus A. Zimmerman 4 52.2575 24 Stefan Larsson 202 52.2334 25 Ben Maloney 1 52.0371 26 Frank Matillek 10 52.0287 27 Wolfgang Schwieger 95 51.9692 28 Marc Wiseler 9 51.8176 29 Diego Bertolotti 1 51.6397 30 Rudi Tapper 4 51.6208 31 Valter Cavecchia 54 51.4133 32 Brian D. Niegocki 26 51.4097 33 Randall Scalise 145 51.3878 34 Hiroyuki Okazaki 55 51.2916 35 Max Dettweiler 33 51.2879 36 Antonio Lucendo 20 51.1797 37 Vaughan Davies 58 51.0924 38 Peter Kaiser 85.3333 51.0002 39 Erik Veit 54 50.9976 40 Alen Kecic 18 50.9744

move up list ↑

move down list ↓

Notes:

Score for Primes

To find the score for a person, program or project's primes, we give each prime n the score (log n)³ log log n; and then find the sum of the scores of their primes. For persons (and for projects), if three go together to find the prime, each gets one-third of the score. Finally we take the log of the resulting sum to narrow the range of the resulting scores. (Throughout this page log is the natural logarithm.)

How did we settle on (log n)³ log log n? For most of the primes on the list the primality testing algorithms take roughly O(log(n)) steps where the steps each take a set number of multiplications. FFT multiplications take about

O( log n ^. log log n ^. log log log n )

operations. However, for practical purposes the O(log log log n) is a constant for this range number (it is the precision of numbers used during the FFT, 64 bits suffices for numbers under about 2,000,000 digits).

Next, by the prime number theorem, the number of integers we must test before finding a prime the size of n is O(log n) (only the constant is effected by prescreening using trial division).  So to get a rough estimate of the amount of time to find a prime the size of n, we just multiply these together and we get

O( (log n)³ log log n ).

Finally, for convenience when we add these scores, we take the log of the result.  This is because log n is roughly 2.3 times the number of digits in the prime n, so (log n)³ is quite large for many of the primes on the list. (The number of decimal digits in n is floor((log n)/(log 10)+1)).